Problem: Given $ \overrightarrow{PQ}\perp\overrightarrow{PS}$, $ m \angle QPR = 7x - 105$, and $ m \angle RPS = 6x - 52$, find $m\angle QPR$. $P$ $Q$ $S$ $R$
Explanation: From the diagram, we see that together ${\angle QPR}$ and ${\angle RPS}$ form ${\angle QPS}$ , so $ {m\angle QPR} + {m\angle RPS} = {m\angle QPS}$ Since we are given that $\overrightarrow{PQ}\perp\overrightarrow{PS}$ , we know ${m\angle QPS = 90}$ Substitute in the expressions that were given for each measure: $ {7x - 105} + {6x - 52} = {90}$ Combine like terms: $ 13x - 157 = 90$ Add $157$ to both sides: $ 13x = 247$ Divide both sides by $13$ to find $x$ $ x = 19$ Substitute $19$ for $x$ in the expression that was given for $m\angle QPR$ $ m\angle QPR = 7({19}) - 105$ Simplify: $ {m\angle QPR = 133 - 105}$ So ${m\angle QPR = 28}$.